Newmark's method to solve a system of spring elements

In summary: At the beginning or at the end of time integration?In summary, the steps to solving the Newmark's method for a system with multiple degrees of freedom are to first define the system, then calculate the displacement, velocity and acceleration for each degree of freedom, and finally combine the results.
  • #1
mina1363
14
0
Hi,

I have been trying to solve a system consisting of 3 linear spring elements using the Newmark's method to find displacement, velocity and acceleration in Matlab. I followed the instructions shown on page 5 of the paper attached and my code does not seem to be working as the displacement values keep increasing. Does anyone know whether the instructions I am using are correct...? Also does anyone know a similar system for which a code has been written already so that I can compare my results.

p.s. I've aslo attached the equation of motionfor a single element. Effectively I have similar system including 4 by 4 mass and stuffness matrices and 4 by 1 displacement and acceleration matrices.

Any hints and tips would be appreciated.

Many Thanks!
 

Attachments

  • Dynamics Analysis by Numercial Integration.pdf
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  • Untitled.jpg
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  • #2
As far as I remember (it was 2 years ago), I relied on the same document and I think that I had the same issue with it.

I have attached a draft of mine that I found were you can have at least another version of the algorithm. Actually this part is based on another thesis.

You can also check Bathe's "Finite Element Procedures" where there's a another version.

Let me know if you still have problems...
 

Attachments

  • chapter.pdf
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  • #3
Thank you for your reply.
I think the steps shown in the document you attached are essentially the same only that they are for a free vibration system with no damping...rather than a forced one...
I'm going to chekc out the other one you mentioned now.
 
  • #4
Hi,
Thank you for mentioning Bathe's book. So I used Newmark's algorithm from :
Bathe, K.-J., (1996), Finite Element Procedures, Prentice-Hall International, page 781 and I did example 9.4 using the simple code I wrote and I got the answers. But I only managed to get the answers because when calculating the effective load, I calculated it for time 't' rather than time 't+deltat' . If I used 't+deltat' I would get really big displacement values. Do you have any idea why this is happening?
 
  • #5
@ mina 1363
hi,i have been doing a similar problem and encountered an exactly similar hurdle.of displacemnts increasing continuoulsy.Can you tell me how you handled this problem
 
  • #6
Hi,

The example in Bathe, K.-J., (1996), Finite Element Procedures is applied to a case with a cosntant force. Unfortuantely I haven't done one for a vairable force yet.

Regards,
 
  • #7
I had a small doubt.Coincidently even I did my initial MATLAB code using the algorithm you used initially ;from the document you have attached.Is there any problem with it?Is it wrong?
 
  • #8
I'm not sure to be honest but at least with the one form the Finite Element Procedures book you can check your algorithm (for a constant force) by doing the example in the book.
If you don't have access to the book try www.scribd.com
 
  • #9
@mina1363
hey thanks a lot!that definitely helped..Seems the initial algorithm has a problem.But I am getting correct value till around 6 "delta t"...not till 12 as given..any clue..why...hey if its possible for you can you send me the MATLAB code you have written..maybe i m somewhere wrong with the code...my email id is... prashastpjain@gmail.com
 
  • #10
Hi, I don't have access to my code right now. If you wish send me your code and I'll have a look at it by all means.
 
  • #11
I m sending you the code in both the ".txt" nd ".m" format...do let me know if there's any error.Your help will mean a lot.
 

Attachments

  • newmark beta method.txt
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  • newmark_B.m
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  • #12
hello mina...

i hvnt heard from you.Any luck with the code?do let me know..
 
  • #13
Hi,

I'm sorry I've not had chance to look at it yet. I will look at it this afternoon and will let you know asap.
 
  • #14
Hi,

I had a look at your code. Everything seems fine. Althought I'm not familiar with the choleski function in Matlab as I just wrote my own algorithm. So make sure that the function gives you L and D. The other thing that I didn't use is the substitution function. Again I'm not sure how that works but make sure that you use backward and forward substitution.

Best of Luck.
 
  • #15
How we can apply Newmark Method for Multiple degrees of freedom
 
  • #16
Hi,
I'm working on "numerical solving the coupled thermoelasticity equation". for time integratio I use newmark method, but I have problem with boundary conditions. Does anyoe know, when should I impose boundary conditions?
 

1. What is Newmark's method used for?

Newmark's method is a numerical integration technique used to solve a system of spring elements. It is often used in structural analysis and mechanical engineering to model the behavior of structures under dynamic loading.

2. How does Newmark's method work?

Newmark's method uses a two-step process to calculate the displacements and velocities of a system of spring elements over time. First, it predicts the values at the next time step using the current values and their derivatives. Then, it corrects these predicted values using a weighted average of the predicted and previous values. This process is repeated until the desired time is reached.

3. What are the advantages of using Newmark's method?

One of the main advantages of Newmark's method is its accuracy in solving systems of spring elements with non-linear behavior. It also allows for larger time steps compared to other methods, which can save computational time. Additionally, Newmark's method can handle both damping and stiffness matrices, making it a versatile method for solving a variety of structural problems.

4. Are there any limitations to using Newmark's method?

Like any numerical method, Newmark's method has its limitations. It may not be suitable for systems with highly non-linear behavior or significant energy dissipation. It also requires the system to have a well-defined stiffness and damping matrix, which may not always be the case in real-world situations.

5. How does Newmark's method compare to other numerical methods?

Newmark's method is a popular choice for solving systems of spring elements because of its accuracy and efficiency. However, it may not always be the best choice for all situations. For example, it may not be as accurate as other methods for highly non-linear systems. Therefore, it is essential to consider the specific characteristics of the problem when choosing a numerical method.

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